I have a simple JDBC connection to Oracle using Oracle's thin JDBC driver which works great.

Oracle Technet has all the docs online for free. You will need to register at technet.oracle.com. Here's the link I found. It spells out how to use the SSL with JDBC Thin drivers.
Oracle calls that package "Oracle Advanced Security" and it has a bunch of info about their hooks into kerberos, and other systems as well as SSL. You will need to enable this feature on your server. In the past (or present) they refered to this feature as NetSSL, but their marketing people and tech people are never on the same page, so expect some confusing docs. Oracle is a full-featured product, but its impossible to configure.